# Involutive distribution

The geometric interpretation of a completely-integrable differential system on an $n$- dimensional differentiable manifold $M ^ {n}$ of class $C ^ {k}$, $k \geq 3$. A $p$- dimensional distribution (or a differential system of dimension $p$) of class $C ^ {r}$, $1 \leq r < k$, on $M ^ {n}$ is a function associating to each point $x \in M ^ {n}$ a $p$- dimensional linear subspace $D( x)$ of the tangent space $T _ {x} ( M ^ {n} )$ such that $x$ has a neighbourhood $U$ with $p$ $C ^ {r}$ vector fields $X _ {1} \dots X _ {p}$ on it for which the vectors $X _ {1} ( y) \dots X _ {p} ( y)$ form a basis of the space $D ( y)$ at each point $y \in U$. The distribution $D$ is said to be involutive if for all points $y \in U$,

$$[ X _ {i} , X _ {j} ] ( y) \in D ( y) ,\ \ 1 \leq i , j \leq p .$$

This condition can also be stated in terms of differential forms. The distribution $D$ is characterized by the fact that

$$D ( y) = \{ {X \in T _ {y} ( M ^ {n} ) } : { \omega ^ \alpha ( y) ( X) = 0 } \} ,\ p < \alpha \leq n ,$$

where $\omega ^ {p+} 1 \dots \omega ^ {n}$ are $1$- forms of class $C ^ {r}$, linearly independent at each point $x \in U$; in other words, $D$ is locally equivalent to the system of differential equations $\omega ^ \alpha = 0$. Then $D$ is an involutive distribution if there exist $1$- forms $\omega _ \beta ^ \alpha$ on $U$ such that

$$d \omega ^ \alpha = \ \sum _ {\beta = p + 1 } ^ { n } \omega ^ \beta \wedge \omega _ \beta ^ \alpha ,$$

that is, the exterior differentials $d \omega ^ \alpha$ belong to the ideal generated by the forms $\omega ^ \beta$.

A distribution $D$ of class $C ^ {r}$ on $M ^ {n}$ is involutive if and only if (as a differential system) it is an integrable system (Frobenius' theorem).

How to Cite This Entry:
Involutive distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involutive_distribution&oldid=47431
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article